Devlin's
Angle

April 2003

The Double Helix

Quick, what do you get when you double a helix?

The answer, as everyone knows, is a Nobel Prize.
Exactly fifty years ago this month, on April 25,
1953, the molecular biologists James D. Watson
and Francis H. C. Crick published their pivotal
paper in Nature in which they described
the geometric shape of DNA, the molecule of life.
The molecule was, they said, in the form of a
double helix - two helices that spiral around
each other, connected by molecular bonds, to
resemble nothing more than a rope ladder that has
been repeatedly twisted along its length. Their
Nobel Prizewinning discovery opened the door to
a new understanding of life in general and
genetics in particular, setting humanity on a
path that in many quite literal ways would change
life forever.

Watson and Crick with their model of DNA (1953),
alongside a modern illustration of the now famous
molecule

"This structure has novel features which are of
considerable biological interest," they wrote.
Well, duh. You're telling me it does. But does
the structure have any mathematical interest? More
generally, never mind the double helix, does the
single helix offer the mathematician much of
interest?

Given the neat way the two intertwined helices
in DNA function in terms of genetic reproduction,
you might think that the helix had important
mathematical properties. But as far as I am aware,
there's relatively little to catch the
mathematician's attention.

The equation of the helix is quite unremarkable. In
terms of a single parameter t, the equation is

x = a cos t, y = a sin t, z = b t

This is simply a circular locus in the xy-plane
subjected to constant growth in the z-direction.

A deeper characterization of a helix is that it is
the unique curve in 3-space for which the ratio of
curvature to torsion is a constant, a result known
as Lancret's Theorem.

Helices are common in the world around us. Various
sea creatures have helical shells, like the ones
shown here

Helical shaped shells

and climbing vines wind around supports to trace out a
helix.

In the technological world of our own making,
spiral staircases, corkscrews, drills, bedsprings, and
telephone handset chords are helix-shaped.

A spiral staircase: where the helix leads to a
higher things

The popular Slinky toy, pictured below, shows that
the helix is capable of providing amusement for even
the most non mathematical among us.

The popular Slinky toy

And what kind of a world would it be without the
binding capacity the helix provides in the form
of various kinds of screws and bolts.

Making a bolt for it: the helix in everyday use

One of the most ingenious uses of a helix was due to
the ancient Greek mathematician Archimedes, who was
born in Syracuse around 287 BC. Among his many
inventions was an elegant device for pumping water
uphill for irrigation purposes. Known nowadays as
the Archimedes screw, it comprised a long,
helix-shaped wooden screw encased in a wooden
cylinder, like this:

The Archimedes screw

By turning the screw, the water is forced up the
tube. The same device was also used to pump water
out of the bilges of ships.

But when you look at each of these useful applications,
you see that there is no deep mathematics involved.
The reason the helix is so useful is that it is the
shape you get when you trace out a circle at the same
time as you move at a constant rate in the
direction perpendicular to the plane of the circle.
In other words, the usefulness of the helix comes
down to that of the circle.

So where does that leave mathematicians as biologists
celebrate the fiftieth anniversary of the discovery
that the helix was fundamental to life? Well, if what
you are looking for is a mathematical explanation of
why nature chose a double helix for DNA, the answer is:
on the sidelines. On this occasion, the mathematics
of the structure simply does not appear to be significant.

On the other hand, that does not mean that Crick and
Watson did not need mathematics to make their discovery.
Quite the contrary. Crick's own work on the x-ray
defraction pattern of a helix was a significant step in
solving the structure of DNA, which involved significant
applications of mathematics (Fourier transforms, Bessel
functions, etc.). Based on these theoretical calculations,
Watson quickly recognized the helical nature of DNA when he
saw one of Rosalind Franklin's x-ray diffraction patterns.
In particular, Watson and Crick looked for parameters
that came from the discrete nature of the DNA helices.

Now, in the scientific advances that followed Crick
and Watson's breakthrough, in particular the cracking
of the DNA code, mathematics was much more to the fore.
But that is another story. In the meantime, I hope I
speak for all mathematicians when I wish the double-helix
a very happy fiftieth birthday.